Try these guys, I think they run a "friendly" Gong show for new acts.
These guys also do a new act/new material night on Thursdays, might be worth getting in touch or popping along and introducing yourself.
Also, if you're willing to travel over to Manchester you've got "Beat the Frog" at the Frog and Bucket and the Gong Show at the Manchester Comedy Store.
Not sure what's running in Chester these days, apart from Alexanders, which is a fully pro night.
Oh and there's also the Comedy Balloon in Manchester, which was always renowned as a nice, friendly gig for newer acts.